Dissertation Defense of CMSE Sunia Tanweer
Department of Computational Mathematics, Science & Engineering
Michigan State University
Dissertation Defense Notice
April 3, 2026 at 10:00am in EB 3540 (ME Seminar Room)
Zoom link: https://msu.zoom.us/j/94126534811
Meeting ID: 941 2653 4811
Passcode: 081752
TITLE: Analyzing Stochastic Time Series and Dynamical Systems with Topology, Stochastic Theory and Machine Learning
Abstract:
In this defense, I will talk about a topological framework for analyzing complex time series arising from dynamical systems, with a particular focus on three fundamental questions: how to distinguish chaos from stochasticity, how to extract meaningful structure from highdimensional signals such as EEG, and how to characterize global phase-space through directed graph homology. I begin by addressing the long-standing problem of distinguishing deterministic chaos from stochastic dynamics using only observed time series. Rather than relying on entropy- or embedding-based methods, I introduce a theoretically grounded approach based on excursion statistics from stochastic process theory. By exploiting universal scaling laws satisfied by continuous semimartingales, I construct a model-free test that can reliably identify diffusiondriven behavior directly from data. Next, I apply topological methods to EEG signals for seizure detection. I use persistence-based features to capture the shape of the signal across scales and combine these features with machine learning models for classification. I explore both singlechannel and multichannel settings, showing that topological summaries can effectively distinguish between preictal, ictal, and interictal brain states, while remaining robust to noise and variability. Finally, I turn to the problem of understanding attractors and basins in dynamical systems. I propose a framework that discretizes the phase space into a Markov chain and uses transition and hitting probabilities to construct graph-based representations of the dynamics. By applying persistent homology to these representations, I show that it is possible to extract qualitative information about basin geometry and distinguish between different types of phase-space organization. Overall, my work demonstrates that topology provides a noise-robust, model-agnostic framework for analyzing stochastic time series and dynamical systems, with applications in healthcare and engineering.
Committee:
Firas A. Khasawneh (chair)
Daniel Segalman
Elizabeth Munch
Hamidreza Modares
Wei-Che Tai
